Deriving the CDF Integral Form of a European Put Price

Summary

The document shows how to rewrite a European put’s discounted expected payoff as an integral of the underlying asset’s cumulative distribution function. Starting from the payoff integral over nonnegative asset prices, it extends the density’s support to the real line and uses the fact that the put payoff is zero when the asset price exceeds the strike. Integration by parts then converts the payoff-weighted density into an integral of the cumulative distribution up to the strike.

The result gives an equivalent representation of the put price and complements the Breeden–Litzenberger relation between strike derivatives of option prices and the risk-neutral density. The derivation assumes a probability density with support on nonnegative prices and vanishing boundary terms. It does not address cases with atoms or other distributional complications, and the source’s original formula uses different dummy variables in its inner and outer integrals.

Key ideas

• A put price is the discounted expectation of its terminal payoff under the relevant pricing distribution.
• The positive-part payoff restricts the density integral to asset prices below the strike.
• Integration by parts turns the payoff-weighted density integral into an integral of the cumulative distribution function.
• The equivalence depends on appropriate support and boundary assumptions.

Tags

# Equivalence of Put Pricing Formulas


# Equivalence of Put Pricing Formulas












I have to show that:

\begin{equation} P_{t,T}(K)=e^{-r(T-t)} \int_0^{\infty}\left(K-S\right)^+ q_T^S(S)dS \end{equation}

is equivalent to: \begin{equation} P_{t,T}(K)=e^{-r(T-t)}\int_{-\infty}^{K}\left(\int_{-\infty}^y q_T^S(z)dz\right)dy \end{equation}

Breeden and Litzenberger have shown that using Leibniz integration rule and differentiating the first equation twice leads to: \begin{equation} q_T^S(K)=e^{rf(T-t)}\frac{\partial^2P_{t,T}(K)}{\partial K^2}\vert_{K=S_T} \end{equation}

However, I have difficulties to directly go from the first to the second equation in an elegant way. Does anyone have an idea how this can be achieved?

Many thanks for the help!

## Answer by RRL (score 6, accepted)

https://quant.stackexchange.com/a/46294

The first equation expresses the option price as a discounted expected value of the payoff contingent on an asset price $S \geqslant 0$. Without loss of generality, we assume that the probability density function has support in $[0,\infty)$, and rewrite as

$$\begin{align} P_{t,T}(K) &=e^{-r(T-t)} \int_{-\infty}^{\infty}\left(K-S\right)^+ q_T^S(S)\,dS \\ &= e^{-r(T-t)} \int_{-\infty}^{K}\left(K-S\right) q_T^S(S)\,dS \end{align} $$

Integrating by parts with $u = K-S$ and $dv = q_T^S(S)\,dS $, we have $du = -dS $ and

$$v = \int_{-\infty}^S q_T^S(z) \, dz,$$

which with vanishing boundaries terms yields the result

$$P_{t,T}(K) = e^{-r(T-t)} \int_{-\infty}^K \left(\int_{-\infty}^Sq_T^S(z) \, dz \right) \, dS$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.